Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2+1679300x+1725369184\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z+1679300xz^2+1725369184z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+2176372125x+80466179063454\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(12916, 1469452)$ | $3.5586912827754635449109631656$ | $\infty$ |
Integral points
\( \left(12916, 1469452\right) \), \( \left(12916, -1482368\right) \)
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-1588062427411286835792$ | = | $-1 \cdot 2^{4} \cdot 3^{4} \cdot 7^{7} \cdot 19 \cdot 23^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{6687833984375}{20278922832} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{9} \cdot 7^{-7} \cdot 19^{-1} \cdot 23 \cdot 53^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7513915036778221237436264324$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.66106202639172232987245787786$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0097890818827624$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.327903043762886$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.5586912827754635449109631656$ |
|
| Real period: | $\Omega$ | ≈ | $0.10592922664435133560437073864$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot1\cdot1\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.5236329854047935939636306393 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.523632985 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.105929 \cdot 3.558691 \cdot 12}{1^2} \\ & \approx 4.523632985\end{aligned}$$
Modular invariants
Modular form 422142.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18794496 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 532 = 2^{2} \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 267 & 2 \\ 267 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 531 & 0 \end{array}\right),\left(\begin{array}{rr} 477 & 2 \\ 477 & 3 \end{array}\right),\left(\begin{array}{rr} 381 & 2 \\ 381 & 3 \end{array}\right),\left(\begin{array}{rr} 531 & 2 \\ 530 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[532])$ is a degree-$11914076160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/532\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 70357 = 7 \cdot 19 \cdot 23^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 140714 = 2 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \) |
| $19$ | split multiplicative | $20$ | \( 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} \) |
| $23$ | additive | $200$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 422142h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 422142j1, its twist by $-23$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.